Will a moving magnetic field exert a force on a stationary charge? An electric charge experiences a force in a magnetic field only when the charge has some velocity and is moving. But velocity is always relative to a given frame of reference. It never exists on its own. If the source of the magnetic field moves, thus moving the magnetic field, and the charge is stationary wrt the earth's surface, the charge is moving in the frame of reference co-moving with the magnetic field's source. Will the electric charge experience a force due to the field in this case?
 A: Yes. A changing magnetic field creates an electric field. This is known as Faraday's law of induction. Induction is the process used in generators to turn mechanical power into electrical power. When induction was first discovered, it was unexpected. However, from a modern perspective, it makes perfect sense. This is because, as you pointed out, the laws of electromagnetism need to be the same in all frames of reference, and this cannot happen without induction.
A: 
If the source of the magnetic field moves, thus moving the magnetic field, and the charge is stationary wrt the earth's surface, the charge is moving in the frame of reference co-moving with the magnetic field's source. Will the electric charge experience a force due to the field in this case?

This is a very astute question. For clarity, let's assume that the charge is mechanically fixed in place relative to the earth and that we can measure the mechanical force on the charge, e.g. with a strain gauge on the mechanical support.
In the source's frame there is a magnetic field, and no electric field. Since the charge is moving in the frame there is a magnetic force on the charge. Due to the magnetic force the charge will push on the support and produce a non-zero strain in the strain gauge.
Now, in the earth's frame the source is moving but the charge is stationary. Since the charge is stationary it will not experience any magnetic force. But as mentioned in the previous paragraph it pushes on the support and produces a non-zero strain in the strain gauge. Since the charge is stationary that force cannot be a magnetic force, so the only possibility is that the force is an electric force which can act on a stationary charge.
This is surprising at first, but makes sense. What is a purely magnetic field in the source's frame is a combination of an electric field and a magnetic field in the earth's frame. Faraday's law says that a changing B field induces a curling E field. In this case the B field is not changing in the source's frame, but it is changing in the earth's frame. This produces the E field which exerts the force on the charge.
So the force on the charge is due to a B field in one frame and due to an E field in another frame. The E field and the B field are not separate entities. They are both parts of one overall electromagnetic field. Different reference frames will decompose that overall electromagnetic field into different electric field and magnetic field components.
